I dropped my iPhone 6 and the screen is a goner. I ordered a replacement but it requires that I mail my broken phone to them after. The thing is is that my phone still works, I can hear my phone receive texts and notifications, but the touch screen is not responsive. Another big thing is that I am unable to connect to iTunes on it as well. Every time I connect it, it continuously connects and disconnects. Is there a way to reset my phone to its factory settings without a working screen and without iTunes? I really do not want to send in my phone with all my personal information and apps on it.
Is it signed in to Find My iPhone? If so you can go to http://icloud.com, select your phone and click erase iPhone.
The good news is that the damage that's preventing you from racing the phone also will prevent someone from connecting it to another iTunes and getting any sort of information from the device.
The level of security that you have depends on the version of iOS on the phone. If you have a passcode and power of the device in the encryption keys will be gone until someone enters the pass code correctly or it's connected to an iTunes computer that the phone already trusts.
The other answers cover how thing should work: connect to iTunes and erase the device using the restore button, logging in to iCloud and sending an erace command to it. I'll add a third option: put the phone in recovery mode and see if it's more stable to allow iTunes the time it needs to erase your data from the device.